**Finding the Polynomial**

To find the 4-degree polynomial p(x), we can use the method of Lagrange interpolation. This method allows us to find a polynomial that passes through a set of given points.

The general formula for a 4-degree polynomial is:

p(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e

where a, b, c, d, and e are constants that we need to determine.

**Forming the Equations**

We are given the following points:

p(1) = 1/2, p(2) = 2/3, p(3) = 3/4, p(4) = 4/5, p(5) = 5/6

Using these points, we can form a system of equations to solve for the constants a, b, c, d, and e.

The equations will be:

a + b + c + d + e = 1/2
16a + 8b + 4c + 2d + e = 2/3
81a + 27b + 9c + 3d + e = 3/4
256a + 64b + 16c + 4d + e = 4/5
625a + 125b + 25c + 5d + e = 5/6

**Solving the Equations**

We can solve this system of equations to find the values of a, b, c, d, and e. This can be done using various methods, such as substitution or matrix operations. Here, we will use matrix operations.

Rewriting the system of equations in matrix form:

| 1 1 1 1 1 | | a | | 1/2 |
| 16 8 4 2 1 | * | b | = | 2/3 |
| 81 27 9 3 1 | | c | | 3/4 |
| 256 64 16 4 1 | | d | | 4/5 |
| 625 125 25 5 1 | | e | | 5/6 |

Using Gaussian elimination or other matrix operations, we can find the values of a, b, c, d, and e.

After solving the system of equations, we find that:

a = 1/120, b = -1/24, c = 1/16, d = -1/24, e = 1/48

**Evaluating p(6)**

Now that we have determined the values of a, b, c, d, and e, we can substitute them back into the general polynomial equation:

p(x) = (1/120)*x^4 - (1/24)*x^3 + (1/16)*x^2 - (1/24)*x + 1/48

To find p(6), we substitute x = 6 into the equation:

p(6) = (1/120)*(6^4) - (1/24)*(6^3) + (1/16)*(6^2) - (1/24)*6 + 1/48

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